Re: Numerus “Numerans-numeratus”

Originally Posted by AndyDora

( classic x/x = isomorphic (x/x_) = x ) for all numbers x

For real numbers, we have $\dfrac{x}{x} = 1$ for $x \neq 0$. You are saying that $\dfrac{x}{x\_} = \underline{x}$. Is it possible you mean $\dfrac{x}{x\_} = \begin{cases}\underline{0} & x=0\\ \underline{1} & \text{otherwise}\end{cases}$? If you meant what you said, that would mean that $\dfrac{5}{5\_} = \underline{5}$, right? I'm afraid that I am getting more confused. I think this math is a bit over my head. I'll stick to the classic math. I cannot figure out how to do simple arithmetic in your system, so I would not be able to use it at all, unfortunately. I appreciate you taking the time to demonstrate it, but I am just not grasping how it all works.

Re: Numerus “Numerans-numeratus”

Originally Posted by SlipEternal

For real numbers, we have $\dfrac{x}{x} = 1$ for $x \neq 0$. You are saying that $\dfrac{x}{x\_} = \underline{x}$. Is it possible you mean $\dfrac{x}{x\_} = \begin{cases}\underline{0} & x=0\\ \underline{1} & \text{otherwise}\end{cases}$? If you meant what you said, that would mean that $\dfrac{5}{5\_} = \underline{5}$, right? I'm afraid that I am getting more confused. I think this math is a bit over my head. I'll stick to the classic math. I cannot figure out how to do simple arithmetic in your system, so I would not be able to use it at all, unfortunately. I appreciate you taking the time to demonstrate it, but I am just not grasping how it all works.

Your assumption to what I meant was entirely correct. I think the difficulty you might be having is entirely my fault.

If you might reconsider your time on this matter....

It was stated in both versions that except regarding the number zero...the entire idea is entirely isomorphic....that should help....

Re: Numerus “Numerans-numeratus”

I clearly messed up when I replied to you Slip...I was reading and replying quickly and thus caused great confusion.
But this is there in the op and the revision...

(x/x_ = 1) for all x =/= 0

(x/0_ = x) for all x

Division is always a (numerical quantity into a dimensional unit quantity)...yielding a (quotient)...
The definition of division never changes...including all cases of zero.

It is the description of numbers and the operations of multiplication and division that change...therefore zero and its operations...but in all cases I have striven to show how this makes no changes whatsoever...except regarding zero.

Re: Numerus “Numerans-numeratus”

I clearly messed up when I replied to you Slip...I was reading and replying quickly and thus caused great confusion.
But this is there in the op and the revision...

(x/x_ = 1) for all x =/= 0

(x/0_ = x) for all x

Division is always a (numerical quantity into a dimensional unit quantity)...yielding a (quotient)...
The definition of division never changes...including all cases of zero.

It is the description of numbers and the operations of multiplication and division that change...therefore zero and its operations...but in all cases I have striven to show how this makes no changes whatsoever...except regarding zero.

Thanks for your time.

I appreciate all of the work you put into this. My advice is to reorganize the work a bit to make it a bit more readable. While your work seems to have developed independently from existing mathematical works on the subject, you might want to explain how your work relates to Wheel Theory or Meadow Theory. Based on what I have seen, it appears that your work might be isomorphic to a meadow where $0^{-1} = 1$. By giving your work context in a larger mathematical framework, it will be easier for mathematicians to interact with your work. Currently, it is very clunky and difficult to work with. Once I know that I can treat it as a meadow for the purposes of arithmetic, it becomes clear that you are attempting to give physical meaning to the fact that you have chosen $0^{-1} = 1$ along with an explanation of why that is a strong choice for the inverse of zero.

But, it appears that you are taking precautions to keep your work separate from existing mathematical knowledge. If this is because you disagree with the basic properties of a meadow or a wheel, then it still may behoove you to introduce your work as a divergence from the current theories, along with an explanation of why those theories fail to satisfy and how your work corrects whatever they get wrong. Once you introduce your work and its interdependencies and divergences from the current mathematical framework, mathematicians will have a clear idea of what your work is meant to accomplish.

That said, you do try to justify your work. You say that the ability to divide by zero should be a boon to physicists and mathematicians alike. However, as someone who has looked into wheel theory and meadow theory (the latter at your suggestion, actually), I find the presentation you have provided vastly more convoluted than the simple presentations of the existing theories. I wonder, why would I choose your theory over the much more intuitive ones I have already read?

Re: Numerus “Numerans-numeratus”

Originally Posted by SlipEternal

I appreciate all of the work you put into this. My advice is to reorganize the work a bit to make it a bit more readable. While your work seems to have developed independently from existing mathematical works on the subject, you might want to explain how your work relates to Wheel Theory or Meadow Theory. Based on what I have seen, it appears that your work might be isomorphic to a meadow where $0^{-1} = 1$. By giving your work context in a larger mathematical framework, it will be easier for mathematicians to interact with your work. Currently, it is very clunky and difficult to work with. Once I know that I can treat it as a meadow for the purposes of arithmetic, it becomes clear that you are attempting to give physical meaning to the fact that you have chosen $0^{-1} = 1$ along with an explanation of why that is a strong choice for the inverse of zero.

But, it appears that you are taking precautions to keep your work separate from existing mathematical knowledge. If this is because you disagree with the basic properties of a meadow or a wheel, then it still may behoove you to introduce your work as a divergence from the current theories, along with an explanation of why those theories fail to satisfy and how your work corrects whatever they get wrong. Once you introduce your work and its interdependencies and divergences from the current mathematical framework, mathematicians will have a clear idea of what your work is meant to accomplish.

That said, you do try to justify your work. You say that the ability to divide by zero should be a boon to physicists and mathematicians alike. However, as someone who has looked into wheel theory and meadow theory (the latter at your suggestion, actually), I find the presentation you have provided vastly more convoluted than the simple presentations of the existing theories. I wonder, why would I choose your theory over the much more intuitive ones I have already read?

This has been very helpful. I will do my best to consider all of your advice. I am very appreciative of this.

If there is a multiplicative inverse for zero....then you violate a field axiom...I understand this is dissolved in meadows by adjusting the field axioms. I do not wish to dissolve any field axioms. The point of this is to show that division by zero is possible period...and is a valid mathematical construct...thus anything more than a field rings/meadows/wheels.... is not even needed. That is why you would want this. Also it as stated it unifies mathematics and physics....which perhaps it is convoluted....time will tell.

It was once convoluted..and pointless to claim the world was round...now it is the exact opposite.

It unifies semantics...and mathematics...and philosophy...that alone is why...the division by zero a by product.

of course these are my opinions....it remains that I am grateful for you advice, help and time.

Re: Numerus “Numerans-numeratus”

Originally Posted by SlipEternal

I appreciate all of the work you put into this. My advice is to reorganize the work a bit to make it a bit more readable. While your work seems to have developed independently from existing mathematical works on the subject, you might want to explain how your work relates to Wheel Theory or Meadow Theory. Based on what I have seen, it appears that your work might be isomorphic to a meadow where $0^{-1} = 1$. By giving your work context in a larger mathematical framework, it will be easier for mathematicians to interact with your work. Currently, it is very clunky and difficult to work with. Once I know that I can treat it as a meadow for the purposes of arithmetic, it becomes clear that you are attempting to give physical meaning to the fact that you have chosen $0^{-1} = 1$ along with an explanation of why that is a strong choice for the inverse of zero.

But, it appears that you are taking precautions to keep your work separate from existing mathematical knowledge. If this is because you disagree with the basic properties of a meadow or a wheel, then it still may behoove you to introduce your work as a divergence from the current theories, along with an explanation of why those theories fail to satisfy and how your work corrects whatever they get wrong. Once you introduce your work and its interdependencies and divergences from the current mathematical framework, mathematicians will have a clear idea of what your work is meant to accomplish.

That said, you do try to justify your work. You say that the ability to divide by zero should be a boon to physicists and mathematicians alike. However, as someone who has looked into wheel theory and meadow theory (the latter at your suggestion, actually), I find the presentation you have provided vastly more convoluted than the simple presentations of the existing theories. I wonder, why would I choose your theory over the much more intuitive ones I have already read?

This has been very helpful. I will do my best to consider all of your advice. I am very appreciative of this.

If there is a multiplicative inverse for zero....then you violate a field axiom...I understand this is addressed in meadows by adjusting the field axioms. I do not wish to dissolve any field axioms. The point of this is to show that division by zero is possible period...and is a valid mathematical construct...thus anything more than a field, rings/meadows/wheels.... is not even needed. That is why you would want this. Also it as stated it unifies mathematics and physics....which perhaps it is convoluted....time will tell.

It was once convoluted..and pointless to claim the world was round...now it is the exact opposite.

It unifies semantics...and mathematics...and philosophy...that alone is why...the division by zero a by product.

of course these are my opinions....it remains that I am grateful for you advice, help and time.

Thank you Slip

If I say...."I have zero money in my account"

What does this statement mean?

It does not mean I have no money...just non in my account.
It does not mean I can not write a check...just that it will bounce.
It does not mean I have no account...just no money...
so what does it mean....

Re: Numerus “Numerans-numeratus”

Originally Posted by AndyDora

This has been very helpful. I will do my best to consider all of your advice. I am very appreciative of this.

If there is a multiplicative inverse for zero....then you violate a field axiom...I understand this is addressed in meadows by adjusting the field axioms. I do not wish to dissolve any field axioms. The point of this is to show that division by zero is possible period...and is a valid mathematical construct...thus anything more than a field, rings/meadows/wheels.... is not even needed. That is why you would want this. Also it as stated it unifies mathematics and physics....which perhaps it is convoluted....time will tell.

It was once convoluted..and pointless to claim the world was round...now it is the exact opposite.

It unifies semantics...and mathematics...and philosophy...that alone is why...the division by zero a by product.

of course these are my opinions....it remains that I am grateful for you advice, help and time.

Thank you Slip

If I say...."I have zero money in my account"

What does this statement mean?

It does not mean I have no money...just non in my account.
It does not mean I can not write a check...just that it will bounce.
It does not mean I have no account...just no money...
so what does it mean....

This is precisely what I mean, and could provide a good introduction to your work. Describing shortcomings you find in mathematics gives a great lead in to what you are trying to fix and why. I cannot say whether other mathematicians will agree with this, but hopefully it will give them a better understanding of why your theory deserves their attention.

Re: Numerus “Numerans-numeratus”

Originally Posted by SlipEternal

This is precisely what I mean, and could provide a good introduction to your work. Describing shortcomings you find in mathematics gives a great lead in to what you are trying to fix and why. I cannot say whether other mathematicians will agree with this, but hopefully it will give them a better understanding of why your theory deserves their attention.

I will consider how to introduce this as the prologue. Thanks for this advice. I have not done so up to now, because I strive to keep it short...for obvious reasons. I do however understand and agree with your point here.

I find that mathematicians hate my example. That they literally hate me...for suggesting it. I am a philosopher. This alone causes them to hate me. No over usage of the word hate here either. This is sad.

Mathematicians and philosophers were born as siblings. They split along time ago. Mathematicians searching only for the concrete. Philosophers searching only for the abstract.

We must unify them...as it should be. Then we will bring back the glorious days of great thought...like we have seen with the ancient mathematicians who where in fact philosophers also.

Re: Numerus “Numerans-numeratus”

I don't recall that anyone has said division by 0 can't be done. You can indeed define it to have a meaning. But it does tend to destroy a few properties of number system you are working in.

-Dan

I understand this. However if it can or can not be done was not what I was referring to with this last link. The NEED for it to be done is now what seems to be the question, and where I seem to be lacking...

I have spent much time working on those "destroyed properties" you mention. I have carefully shown in the revised and op editions (I hope) how all field axioms remain unchanged...except regarding zero (and even then only the identity changes, and infinitesimally so). Perhaps there is yet an error....but....I still think it can be done without "touching" a single axiom...as suggested...and so far shown (again I hope).

Are you of the accord that even if I could do so...such as in meadows and wheels..that it would still remain pointless?
I hope to add here a collection of examples where there is a point....after I have amassed and considered them deeply.